A sandwich with segment convexity

The aim of this note is to give a sufficient condition for pairs of functions to have a convex separator when the underlying structure is a Cartan--Hadamard manifold, or more generally: a reduced Birkhoff--Beatley system. Some exotic behavior of convex hulls are also studied.


INTRODUCTION
As it is well-known, separation theorems play a crucial role in many fields of Analysis and Geometry, and they can be interesting on their own right. Let us quote here the convex separation theorem of Baron, Matkowski, and Nikodem [1], one of our main motivations: Theorem. Let D be a convex subset of a real vector space X, and let f, g : D → R be given functions. There exists a convex separator for f and g if and only if holds for all n ∈ N, x 0 , . . . , x n ∈ D and t 0 , . . . , t n ∈ [0, 1] with t 0 +· · ·+t n = 1. Moreover, if X is finite dimensional, then the length of the involved combinations can be restricted to n ≤ dim(X).
In this challenge, one has to face two crucial problems. Firstly: What kind of structures should be used to have convexity without convex combinations? Secondly: What is the corresponding form of (1) in lack of algebraic manipulations? The proper choice to the structure turns out to be Birkhoff-Beatley systems, the generalizations of Cartan-Hadamard manifolds. Inequality (1) has to be replaced by another one, in order that an iteration process can be applied.

CONVEX SEPARATION IN BIRKHOFF-BEATLEY SYSTEMS
The precise axiomatic discussion of Euclidean geometry is due to Hilbert [7]; a nice and simplified presentation can be found in the book of Hartshorne [6]. Later Birkhoff initiated [3] and then together with Beatley elaborated [4] an elegant and didactic approach which is based on the ruler and the protractor. In what follows, we shall need some of their notions and axioms.
Assume that X is the set of points with at least two elements. Consider a family L of subsets of X whose elements are termed lines. Let further d : X 2 → R be a given function called a metric. We require two axioms: The postulate of incidence and the postulate of the ruler: • Any two distinct points determine a unique line containing them.
• For each ∈ L there exists a bijection c : R → such that d c(t), c(s) = |t − s|.
In this case, we say that (X, L , d) is a reduced Birkhoff-Beatley system. A bijection c : R → satisfying the condition in the second postulate is called a ruler for . The postulate of the ruler implies immediately, that each line has at least two points. Moreover, we can introduce a ternary relation called betweenness on X: the point b is between the points a and c if a, b, c are three different collinear points, and d(a, c) = d(a, b) + d(b, c). Using the abbreviation (abc) to this fact, one can prove that the axioms of abstract betweenness are satisfied: • If (abc), then a, b, c are pairwise distinct and collinear; further, (cba).
• For distinct points a, b, there exists c such that (abc).
Using betweenness, the notion of line segment [a, b] spanned by the points a, b can be defined in the following way. If a = b, then [a, b] := {a}; otherwise, If a = b, and is the unique line passing through them, then let c : R → be a ruler for such that c(α) = a and c(β) = b. Then we call the bijectioñ the standard parametrization of the segment [a, b]. Clearly,c(0) = a, andc(1) = b. When there is no risk of confusion, we shall also denote a standard parametrization simply by c, without tilde. Note that, unlike a ruler, a standard parametrization does not need to be distance preserving (unless d(a, b) = 1).
Once having segments, we have convexity concepts. A set K ⊆ X is convex if [a, b] ⊆ K holds for all a, b ∈ K. The family of convex sets is denoted by C (X). It turns out that C (X) is a convex structure indeed in the abstract sense of van de Vel [14]. That is, • X and ∅ are convex sets; • the intersection of convex sets is convex; • the union of nested convex sets is convex.
The convex hull of H ⊆ X, as usual, is the smallest convex set that contains H: It can be proved that segments are convex, and a set H is convex if and only if conv(H) = H. Moreover, the mapping conv : P(X) → P(X) is a hull operator, that is, an idempotent, monotone and extensive map. For further precise details, we refer to the paper [2] or to the excellent monograph [14]. Convex hulls are finitely inner represented in the following sense: is a reduced Birkhoff-Beatley system and H ⊆ X, then Proof. Denote the right-hand side of the formula above by K. Then H ⊆ K holds evidently; moreover, for each finite set F ⊆ H, we have that conv(F ) ⊆ conv(H). Thus K ⊆ conv(H). To complete the proof we have to show that K is convex. If a, b ∈ K, then a ∈ conv(F a ) and b ∈ conv(F b ) with suitable finite subsets F a and F b of H.
Note, that convex hulls are finitely inner represented in any convex structure. The proof of this fact is based on transfinite methods, and can be found in the monograph [14]. When convexity is defined via segments, the presented elementary approach can also be followed.
Unfortunately, neither the definition, nor Lemma 1 provides a constructive method for finding the convex hull of a concrete set. Especially for finite sets, the formula of Lemma 1 terminates in a 'circulus vitiosus'. Therefore the fixed point theorem of Kantorovich [9] will play a distinguished role for us. Its iteration process is a constructive method to approximate convex hulls. Proof. Clearly, L := {H n | n ∈ N} is an increasing chain, and H ⊆ L . The iteration process guarantees that L is a convex set. Let C ⊆ X be a convex set such that H 1 = H ⊆ C. Then H 2 ⊆ C by the convexity of C. By induction, H n ⊆ C for all n ∈ N. Thus L ⊆ C. In other words, L is the smallest convex set containing H, and the proof is completed.
Assume that (X, L , d) is a reduced Birkhoff-Beatley system, and let X * := X × R. For arbitrary elements (x 0 , y 0 ) and (x 1 , y 1 ) of X * , the first projections determine a line in L provided that x 0 = x 1 . Let c : R → be a ruler for such that c(s 0 ) = x 0 and c(s 1 ) = x 1 hold, and define Then * := {c * (t) | t ∈ R} is called the line connecting (x 0 , y 0 ) and (x 1 , y 1 ). If x 0 = x 1 , and y 0 = y 1 , then let c : R → X be the constant mapping given by c(t) = x 0 , and then define the line connecting (x 0 , y 0 ) and (x 1 , y 1 ) by the same formulae as above. In this way, we can specify the lines of X * denoted by L * . Finally, define the metric on X * by The triple (X * , L * , d * ) obtained in this way will be called the vertical extension of the reduced Birkhoff-Beatley system (X, L , d). The most important property of vertical extensions is subsumed by the following lemma. Proof. It can easily be checked that the postulate of incidence is valid in the vertical extension.
Keeping the previous notations, consider a line * determined by (x 0 , y 0 ) and (x 1 , y 1 ) with distinct first projections. We claim that c * serves as a ruler for * . Indeed, since c is a ruler for , we arrive at

the case of vertical lines, can be handled similarly.
Assume that D is a nonempty convex subset in a reduced Birkhoff-Beatley system. We say that a function ϕ : D → R is segment convex, or simply: convex, if Our main result gives a sufficient condition for the existence of a convex separator. To formulate it, we need the following concept. We say that a reduced Birkhoff-Beatley system (X, L , d) is drop complete if, for each convex set K ⊆ X and for all x 0 ∈ X, the usual drop representation holds: conv Theorem 1. Let D be a convex set in a reduced Birkhoff-Beatley system (X, L , d) whose vertical extension is drop complete. If, for all n ∈ N, x 0 , . . . , x n ∈ D, x ∈ conv{x 1 , . . . , x n }, and for all t ∈ [0, 1], the functions f, g : D → R satisfy the inequality where c : [0, 1] → D is the segment joining x 0 = c(0) and x = c(1), then there exists a convex function ϕ : Proof. Let E := conv(epi(g)). First we show, that f (x) ≤ y whenever (x, y) ∈ E. By Lemma 1, each point of E belongs to the convex hull of some finite subset of epi(g). If (x, y) belongs to a singleton, then f (x) ≤ y holds trivially. Assume that the desired inequality holds if (x, y) belongs to the convex hull of any n element subset of epi(g). Consider the case when The vertical extension is a drop complete reduced Birkhoff-Beatley system, thus there exists a point (x * , y * ) and t ∈ [0, 1] such that where c : [0, 1] → D is the segment fulfilling c(0) = x 0 and c(1) = x * . Using the inductive assumption and (2), we arrive at which was our claim. This property ensures that the formula Taking the infimum at y 0 and y 1 , we get the convexity of ϕ, and this completes the proof.
Let H be an arbitrary subset of a reduced Birkhoff-Beatly system, and assume that for each x ∈ conv(H) there exists F ⊆ H such that x ∈ conv(F ) and card(F ) ≤ κ. The least possible κ with this property is called the Carathéodory number of the system. If there does not exist such a κ, then the Carathéodory number is defined to be +∞. Equivalently, the representation of Lemma 1 remains valid if we allow only the convex hulls of sets with at most κ elements on the right-hand side. If the Carathédory number of the vertical extension is known, we can strengthen the statement of Theorem 1. The proof is essentially the same, thus we omit it.
Theorem 2. Keeping the conditions of the previous theorem, assume that the Carathéodory number of the vertical extension is κ. Then the size of the involved convex hull can be reduced to n ≤ κ.
Assume that the underlying reduced Birkhoff-Beatly system is a vector space. Then, using induction, one can check easily that (2) implies (1). Unfortunately, the converse implication is not valid. Thus our main results are only sufficient conditions for the existence of a convex separator. However, under this generality, a full characterization cannot be expected.
If X is a finite dimensional vector space, the classical separation result of [1] restricts the length of the involved combination to dim(X)+1, while the Carthéodory number of the vertical extension is κ = dim(X) + 2. In other words, the reduction of Theorem 2 can be improved in the finite dimensional setting.
The drop completeness of the vertical extensions is required both in Theorem 1 and Theorem 2. Clearly, this assumption makes the underlying reduced Birkhoff-Beatley system drop complete as well. Thus the question arises, quite evidently: Does the extension inherit drop completeness from the original system? In order to justify the conditions of the main results, we will give a negative answer in the last section.

CONVEX SEPARATION IN CARTAN-HADAMARD MANIFOLDS
The celebrated theorem of Hopf states that each simply connected, complete Riemannian manifold of positive sectional curvature is compact. In contrast to this behavior, nonpositive curvature results in an opposite feature according to the theorem of Cartan and Hadamard: Theorem. The exponential map at any point of a simply connected, complete d dimensional Riemannian manifold of nonpositive sectional curvature is a global diffeomorphism between R d and the manifold.
These manifolds are called Cartan-Hadamard manifolds. In particular, by this theorem, each Cartan-Hadamard manifold is homeomorphic to a Euclidean space. Moreover, geodesics can be parametrized along the entire set of reals, and two geodesics can have at most one common point. This means that the postulate of incidence and the postulate of ruler are satisfied, and we can formulate the next statement.

Lemma 4. Each Cartan-Hadamard manifold is a reduced Birkhoff-Beatley system.
By Lemma 3 and Lemma 4, the vertical extension of a Cartan-Hadamard manifold is a reduced Birkhoff-Beatley system. Moreover, now the vertical extension has a very close relation with the product Riemannian metric. In fact, exactly this relation (formulated in the next lemma) has motivated the notion of vertical extensions. For the technical background of the proof, we refer to the monograph of Sakai [12]. Clearly, the product manifold is a simply connected and complete Riemannian manifold. Moreover, using the Koszul formulae for the Christoffel symbols of M × R, we can conclude that Γ k ij = 0 whenever (d + 1) ∈ {i, j, k}. Consider now a geodesic c * in the product manifold M × R. Since its coordinate functions satisfy the second-order differential equations c * k + Γ k ij • c * c * i c * j = 0 and the Christoffel symbols have the previously mentioned properties, c * (d+1) = 0 follows. Thus the last component of c * is affine. Furthermore, by the behavior of the Christoffel symbols and by the geodesic differential equation again, the first projection of c * results in a geodesic of M . Therefore any geodesic c * connecting the points (x 0 , y 0 ) and (x 1 , y 1 ) of the product manifold M × R can be parametrized as where the geodesic c of M connects the points c(0) = x 0 and c(1) = x 1 . Note also that this parametrization is a global one in case of Cartan-Hadamard manifolds. This shows that the product manifold M × R is the vertical extension of M .
In particular, the vertical extension is a simply connected and complete manifold, as well. Now we show that its sectional curvature is nonpositive. Let σ ⊂ T (x,y) (M × R) = T x M ⊕ R be an arbitrary plane, and let b 1 , b 2 be a base in σ.
which is a contradiction. Thus dim(T x M ∩ σ) ≥ 1, and we may assume that the second direct component of b 2 is zero. Then b 1 has a direct decomposition b 1 = b 11 + b 12 . Recall that the sign of the sectional curvature depends only on the sign of R(b 1 , b 2 , b 2 , b 1 ), where R is the Riemannian curvature tensor. Since its components can be obtained by we arrive at R ijkl = 0 provided that (d + 1) ∈ {i, j, k, l}. Thus R vanishes if one of the arguments contains b 12 . Therefore, since the second direct components of b 11 and b 2 are zero and the sectional curvature of M is nonpositive. This completes the proof.
For more details on Cartan-Hadamard manifolds, we recommend the book of Jost [8]. As direct consequences of Theorem 1 and Theorem 2, we can formulate the next two corollaries. Corollary 1. Let D be a convex set in a Cartan-Hadamard manifold M whose vertical extension is drop complete. If, for all n ∈ N, x 0 , . . . , x n ∈ D, x ∈ conv{x 1 , . . . , x n }, and for all t ∈ [0, 1], the functions f, g : D → R satisfy (2) where c : [0, 1] → R is the geodesic segment joining x 0 = c(0) and x = c(1), then there exists a convex function ϕ : D → R fulfilling f ≤ ϕ ≤ g.

Corollary 2.
Keeping the conditions of the previous theorem, assume that the Carathéodory number of the vertical extension is κ. Then the size of the involved convex hull can be reduced to n ≤ κ.
Cartan-Hadamard manifolds and Euclidean spaces are quite "close" relatives. Hence one may expect that the Carathéodory number of a Cartan-Hadamard manifold M , accordingly to the Euclidean case, can be expressed as κ = dim(M ) + 1. However, as far as we know, this is still an open problem posed by Ledyaev, Treiman, and Zhu [10].

THE EXOTIC BEHAVIOR OF CONVEX HULLS
The aim of this section is to prove, that drop completeness of the vertical extension cannot be changed to drop completeness of the underlying system in Theorem 1 and Theorem 2. The main reason is the exotic behavior of convex hulls: It may occur that the convex hull of three points in a Cartan-Hadamard manifold is not contained in a two dimensional submanifold. To construct such an example, let us recall here some basic facts in hyperbolic geometry. For references, see the book of Ratcliffe [11].
The hyperbolic plane, denoted by H 2 in the forthcomings, is a two dimensional Cartan-Hadamard manifold with constant sectional curvature −1. We will use two of its several models. The first one is the Beltrami-Klein model (known also as the Cayley-Klein model): Here the plane is the open unit disc, and the lines are its Euclidean chord segments. The distance of this model will not be used.
The second model is the Poincaré half-plane model. The plane is the upper open Cartesian half-plane R × R + ; lines are either circles with center on the boundary line or vertical Euclidean half-lines. Its metric plays a key role in our investigation. The distance of a = (a 1 , a 2 ) and b = (b 1 , b 2 ) is given by In particular, if a 1 = b 1 , this formula can be simplified (which will be quite convenient for us): If Since geodesics are of constant speed, we have d(x 0 , x) = td(x 0 , x 1 ) for all t ∈ [0, 1], where x = c(t). Thus we can reconstruct the points (x, y) between (x 0 , y 0 ) and (x 1 , y 1 ) from x as Theorem 3. The hyperbolic plane is drop complete, while its vertical extension is not. Proof.
The Beltrami-Klein model shows, that the convex structure of H 2 can be identified with the convex structure of the open disc inherited from R 2 . In particular, the drop representation is valid in H 2 . Now we prove, that the vertical extension H 2 × R is not drop complete. We will illustrate it using the convex hull of the points Thus the second common projection of P and Q is obtained via (5) as The estimation above can be checked even by hand. Moreover, the points of the segment [P, Q] share this common last component. Therefore, Clearly, R = ((0, √ 41), 1) ∈ [B, C]. Now we determine that point of the vertical extension, whose first projection is x, and belongs to the segment [A, R]. Using (4) and (5) This estimation, with a bit more effort, can also be checked by hand. Thus, Since X 1 = X 2 , we can conclude that the drop representation involving {A} and [B, C] does not cover the entire convex hull of A, B, C, which was to be proved.
The Beltrami-Klein open sphere model and the first part of the argument show, that the geodesic convex structure of hyperbolic space is compatible with the Euclidean convex sturcture of the open ball. In particular, the hyperbolic space is drop complete in any dimension, and its combinatorial invariants coincide with the standard Euclidean ones. Using the Cartan-Hadamard theorem (or the results of [2], it can be proved that these properties are also true for two dimensional Cartan-Hadamard manifolds.
As we have already mentioned, the greatest advantage of Lemma 2 is that it can be implemented. In fact, the theorem above was conjectured via a computer algorithm. In what follows, we sketch breafly its pseudo code.
We shall need two functions. The first one, geod calculates a point of a geodesic between two points: geod : (H 2 × R) × (H 2 × R) × R → H 2 × R so that geod(A, B, 0) = A and geod(A, B, 1) = B hold. The function dist calculates the hyperbolic distance of two points in H 2 : dist(a, b) : H 2 × H 2 → R.
The list ConvexHull collects the points of the convex hull as an ordered list. Initially we put the points of the set whose convex hull is to be computed into ConvexHull. ConvexHull[i] is the ith element in ConvexHull. Indexing starts with 0. The parameter iterations is the number of iterations. Finally, the parameter res is the hyperbolic distance between points to be calculated along geodesics. The algorithm takes two points A and B from ConvexHull and adds the points of the geodesic from A to B with hyperbolic distance res from each other to ConvexHull. The point B is chosen so that repetitions are avoided.
• The variables size and previoussize keep track of the number of points calculated in the convex hull. Initially size := |ConvexHull|.
previoussize := 0. • The following loop is to be performed iterations many times.
-For i = 0, . . . , size, perform the following: * For j = max(previoussize, i + 1), . . . , size, perform the following: Using our algorithm, we can illustrate some points of conv{A, B, C} in the proof of Theorem 3. The first figure is the intersection of the approximation of the convex hull with the plane [a, r] × R: The second figure is the intersection of the approximation with [p, q] × R: In each case, iterations = 2 and res = 0.006. In the concrete implementation, points are plotted whose distance from the planes is at most 0.01.
The program is available and freely downloadable from the homepage below: http://shrek.unideb.hu/˜ftzydk/convex/